A069905 -id:A069905 - cp's OEIS frontend

A344294 5-smooth but not 3-smooth numbers k such that A056239(k) >= 2*A001222(k).

5, 10, 15, 25, 30, 45, 50, 75, 90, 100, 125, 135, 150, 225, 250, 270, 300, 375, 405, 450, 500, 625, 675, 750, 810, 900, 1000, 1125, 1215, 1250, 1350, 1500, 1875, 2025, 2250, 2430, 2500, 2700, 3000, 3125, 3375, 3645, 3750, 4050, 4500, 5000, 5625, 6075, 6250

Offset: 1

Gus Wiseman, May 16 2021

nonn

A number is d-smooth iff its prime divisors are all <= d.

A prime index of k is a number m such that prime(m) divides k, and the multiset of prime indices of k is row k of A112798. This row has length A001222(k) and sum A056239(k).

			The sequence of terms together with their prime indices begins:
       5: {3}           270: {1,2,2,2,3}
      10: {1,3}         300: {1,1,2,3,3}
      15: {2,3}         375: {2,3,3,3}
      25: {3,3}         405: {2,2,2,2,3}
      30: {1,2,3}       450: {1,2,2,3,3}
      45: {2,2,3}       500: {1,1,3,3,3}
      50: {1,3,3}       625: {3,3,3,3}
      75: {2,3,3}       675: {2,2,2,3,3}
      90: {1,2,2,3}     750: {1,2,3,3,3}
     100: {1,1,3,3}     810: {1,2,2,2,2,3}
     125: {3,3,3}       900: {1,1,2,2,3,3}
     135: {2,2,2,3}    1000: {1,1,1,3,3,3}
     150: {1,2,3,3}    1125: {2,2,3,3,3}
     225: {2,2,3,3}    1215: {2,2,2,2,2,3}
     250: {1,3,3,3}    1250: {1,3,3,3,3}

Allowing any number of parts and sum gives A080193, counted by A069905.
The partitions with these Heinz numbers are counted by A325691.
Relaxing the smoothness conditions gives A344291, counted by A110618.
Allowing 3-smoothness gives A344293, counted by A266755.
A025065 counts partitions of n with at least n/2 parts, ranked by A344296.
A035363 counts partitions of n whose length is n/2, ranked by A340387.
A051037 lists 5-smooth numbers (complement: A279622).
A056239 adds up prime indices, row sums of A112798.
A257993 gives the least gap of the partition with Heinz number n.
A300061 lists numbers with even sum of prime indices (5-smooth: A344297).
A342050/A342051 list Heinz numbers of partitions with even/odd least gap.
Cf. A000041, A001399, A002865, A027336, A244990, A261144, A344295.

Select[Range[1000],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&&Max@@First/@FactorInteger[#]==5&]

Intersection of A080193 and A344291.

A014591 a(n) = floor(n^2/12 + 5/4).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 17, 20, 22, 25, 28, 31, 34, 38, 41, 45, 49, 53, 57, 62, 66, 71, 76, 81, 86, 92, 97, 103, 109, 115, 121, 128, 134, 141, 148, 155, 162, 170, 177, 185, 193, 201, 209, 218, 226, 235, 244, 253, 262, 272, 281, 291, 301, 311, 321

Offset: 0

David Broadhurst

Number of partitions of n + 10 into 4 distinct parts one of which is 3. - Michael Somos, Nov 03 2011

Number of partitions of n into 3 or fewer distinct parts. - Mo Li, Sep 27 2019

			1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 8*x^9 + ...
10 = 4 + 3 + 2 + 1, 11 = 5 + 3 + 2 + 1, 12 = 6 + 3 + 2 + 1, 13 = 7 + 3 + 2 + 1 = 5 + 4 + 3 + 1, 14 = 8 + 3 + 2 + 1 = 5 + 4 + 3 + 2, 15 = 9 + 3 + 2 + 1 = 6 + 5 + 3 + 1 = 6 + 4 + 3 + 2.

Jan Kneissler, The number of primitive Vassiliev invariants of degree up to 12, arXiv:q-alg/9706022, 1997.

It may be only a coincidence that the first 11 terms reproduce all available data on Vassiliev invariants from diagrams with u=2 univalent vertices, as recorded in the Kneissler paper.

Floor[Range[0,70]^2/12+5/4] (* Harvey P. Dale, Oct 22 2013 *)
Table[Length[Select[IntegerPartitions[k, 3], DuplicateFreeQ]], {k,1,50}] (* Mo Li, Sep 27 2019 *)

{a(n) = (n^2 + 3) \ 12 + 1} /* Michael Somos, Nov 03 2011 */

G.f.: (1/(1-x^3)-x^2)/(1-x)/(1-x^2).

a(-n) = a(n). a(n) = 1 + A069905(n). - Michael Somos, Nov 03 2011

More terms from Erich Friedman

A060022 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 3.

Original entry on oeis.org

1, 0, 1, 0, 0, -1, -1, -3, -3, -5, -6, -8, -9, -12, -13, -16, -18, -21, -23, -27, -29, -33, -36, -40, -43, -48, -51, -56, -60, -65, -69, -75, -79, -85, -90, -96, -101, -108, -113, -120, -126, -133, -139, -147, -153, -161, -168, -176, -183, -192, -199, -208, -216, -225, -233, -243, -251, -261, -270, -280

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference between the number of partitions of n+2 into 2 parts and the number of partitions of n+2 into 3 parts. - Wesley Ivan Hurt, Apr 16 2019

Colin Barker, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A004526, A069905.

Cf. For other values of N: this sequence (N=3), A060023 (N=4), A060024 (N=5), A060025 (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

Vec((1 - x - x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)) + O(x^40)) \\ Colin Barker, Apr 17 2019

a(n) = A004526(n+2) - A069905(n+2). - Wesley Ivan Hurt, Apr 16 2019
From Colin Barker, Apr 17 2019: (Start)
G.f.: (1 - x - x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n>5.
(End)

A060023 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 4.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, -1, -1, -3, -4, -7, -8, -13, -15, -20, -24, -31, -35, -44, -50, -60, -68, -80, -89, -104, -115, -131, -145, -164, -179, -201, -219, -243, -264, -291, -314, -345, -371, -404, -434, -471, -503, -544, -580, -624, -664, -712, -755, -808, -855, -911, -963, -1024, -1079, -1145

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+3 into 3 parts and the number of partitions of n+3 into 4 parts. - Wesley Ivan Hurt, Apr 16 2019

Colin Barker, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -2, 0, 0, 1, 1, -1).

Cf. A026810, A069905.

Cf. For other values of N: A060022 (N=3), this sequence (N=4), A060024 (N=5), A060025 (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

I:=[1,0,1,1,1,0,1,-1,-1,-3]; [n le 10 select I[n] else Self(n-1)+Self(n-2)-2*Self(n-5)+Self(n-8)+Self(n-9)-Self(n-10): n in [1..60]]; // Vincenzo Librandi, Jun 23 2015

CoefficientList[Series[(1-x-x^4)/Times@@(1-x^Range[4]),{x,0,60}],x] (* or *) LinearRecurrence[{1,1,0,0,-2,0,0,1,1,-1},{1,0,1,1,1,0,1,-1,-1,-3},70] (* Harvey P. Dale, Jan 14 2015 *)

Vec((1 - x - x^4) / ((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^40)) \\ Colin Barker, Apr 17 2019

a(n) = A069905(n+3) - A026810(n+3). - Wesley Ivan Hurt, Apr 16 2019
From Colin Barker, Apr 17 2019: (Start)
G.f.: (1 - x - x^4) / ((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n>9.
(End)

A325688 Number of length-3 compositions of n such that every distinct consecutive subsequence has a different sum.

Original entry on oeis.org

0, 0, 0, 1, 0, 4, 5, 12, 12, 25, 24, 40, 41, 60, 60, 85, 84, 112, 113, 144, 144, 181, 180, 220, 221, 264, 264, 313, 312, 364, 365, 420, 420, 481, 480, 544, 545, 612, 612, 685, 684, 760, 761, 840, 840, 925, 924, 1012, 1013, 1104, 1104, 1201, 1200, 1300, 1301, 1404

Offset: 0

Gus Wiseman, May 15 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

Confirmed recurrence relation from Colin Barker for n <= 5000. - Fausto A. C. Cariboni, Feb 13 2022

			The a(3) = 1 through a(8) = 12 compositions:
  (111)  (113)  (114)  (115)  (116)
         (122)  (132)  (124)  (125)
         (221)  (222)  (133)  (143)
         (311)  (231)  (142)  (152)
                (411)  (214)  (215)
                       (223)  (233)
                       (241)  (251)
                       (322)  (332)
                       (331)  (341)
                       (412)  (512)
                       (421)  (521)
                       (511)  (611)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..5000

Column k = 3 of A325687.
Cf. A000217 (all length-3).
Cf. A000079, A001399, A005044, A008642, A069905, A108917, A124278, A266223.
Cf. A325686, A325689, A325690, A325691.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],UnsameQ@@Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]&]],{n,0,30}]

Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: x^3*(1 + 2*x^2 + 4*x^3 + 5*x^4) / ((1 - x)^3*(1 + x)^2*(1 + x + x^2)).
a(n) = 2*a(n-2) + a(n-3) - a(n-4) - 2*a(n-5) + a(n-7) for n>7.
(End)

A325690 Number of length-3 integer partitions of n whose largest part is not the sum of the other two.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 2, 4, 3, 7, 6, 10, 9, 14, 13, 19, 17, 24, 23, 30, 28, 37, 35, 44, 42, 52, 50, 61, 58, 70, 68, 80, 77, 91, 88, 102, 99, 114, 111, 127, 123, 140, 137, 154, 150, 169, 165, 184, 180, 200, 196, 217, 212, 234, 230, 252, 247, 271, 266, 290, 285, 310

Offset: 0

Gus Wiseman, May 15 2019

nonn

Confirmed recurrence relation from Colin Barker for n <= 10000. - Fausto A. C. Cariboni, Feb 19 2022

			The a(3) = 1 through a(13) = 14 partitions (A = 10, B = 11):
  (111)  (221)  (222)  (322)  (332)  (333)  (433)  (443)  (444)   (544)
         (311)  (411)  (331)  (521)  (432)  (442)  (533)  (543)   (553)
                       (421)  (611)  (441)  (622)  (542)  (552)   (643)
                       (511)         (522)  (631)  (551)  (732)   (652)
                                     (531)  (721)  (632)  (741)   (661)
                                     (621)  (811)  (641)  (822)   (733)
                                     (711)         (722)  (831)   (742)
                                                   (731)  (921)   (751)
                                                   (821)  (A11)   (832)
                                                   (911)          (841)
                                                                  (922)
                                                                  (931)
                                                                  (A21)
                                                                  (B11)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

Column k = 3 of A325592.
Cf. A000041, A001399, A005044, A008642, A069905, A266223.
Cf. A325689, A325691, A325694.

Table[Length[Select[IntegerPartitions[n,{3}],#[[1]]!=#[[2]]+#[[3]]&]],{n,0,30}]

Conjectures from Colin Barker, May 15 2019: (Start)
G.f.: x^3*(1 + x^2 + x^3 + x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>8.
(End)

A360010 First part of the n-th weakly decreasing triple of positive integers sorted lexicographically. Each n > 0 is repeated A000217(n) times.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8

Offset: 1

Gus Wiseman, Feb 11 2023

nonn

			Triples begin: (1,1,1), (2,1,1), (2,2,1), (2,2,2), (3,1,1), (3,2,1), (3,2,2), (3,3,1), (3,3,2), (3,3,3), ...

For pairs instead of triples we have A002024.
Positions of first appearances are A050407(n+2) = A000292(n)+1.
The zero-based version is A056556.
The triples have sums A070770.
The second instead of first part is A194848.
The third instead of first part is A333516.
Concatenating all the triples gives A360240.
Cf. A000217, A003056, A056557, A056558, A069905, A158842, A331195.

nn=9;First/@Select[Tuples[Range[nn],3],GreaterEqual@@#&]

from math import comb
from sympy import integer_nthroot
def A360010(n): return (m:=integer_nthroot(6*n,3)[0])+(n>comb(m+2,3)) # Chai Wah Wu, Nov 04 2024

a(n) = A056556(n) + 1 = A331195(3n) + 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + log(2)/4. - Amiram Eldar, Feb 18 2024
a(n) = m+1 if n>binomial(m+2,3) and a(n) = m otherwise where m = floor((6n)^(1/3)). - Chai Wah Wu, Nov 04 2024

A375580 a(n) is the number of partitions n = x + y + z of positive integers such that xyz is a perfect cube.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 3, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 0, 2, 2, 2, 2, 1, 2, 3, 2, 2, 3, 2, 0, 1, 1, 3, 1, 3, 2, 2, 1, 1, 1, 2, 1, 4, 1, 2, 3, 3, 3, 3, 1, 1, 4, 2, 2, 2, 3, 1, 2, 3, 1, 3, 4, 1, 3, 2, 2, 1, 2, 2, 3, 3, 2, 4

Offset: 0

Felix Huber, Aug 19 2024

nonn

a(n) is also the number of distinct integer-sided cuboids with total edge length 4*n whose unit cubes can be grouped to a cube.

Conjecture: for n > 176, a(n) > 0. - Charles R Greathouse IV, Aug 20 2024

			a(21) = 3 because the three partitions [1, 4, 16], [3, 6, 12], [7, 7, 7] satisfy the conditions: 1 + 4 + 16 = 21 and 1*4*16 = 4^3, 3 + 6 + 12 = 21 and 3*6*12 = 6^3, 7 + 7 + 7 = 21 and 7*7*7 = 7^3.
See also linked Maple code.

Felix Huber and Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (up to 1000 from Huber)
Felix Huber, Maple codes
David A. Corneth, PARI program

Cf. A338939, A375512, A375576, A103277, A069905.

Maple
```
# See Huber link.
```

a(n)=sum(x=1,n\3, sum(y=x,(n-x)\2, ispower(x*y*(n-x-y),3))) \\ Charles R Greathouse IV, Aug 20 2024

PARI
```
\\ See Corneth link
```

from sympy import integer_nthroot
def A375580(n): return sum(1 for x in range(n//3) for y in range(x,n-x-1>>1) if integer_nthroot((n-x-y-2)*(x+1)*(y+1),3)[1]) # Chai Wah Wu, Aug 21 2024

Trivial upper bound: a(n) <= A069905(n). - Charles R Greathouse IV, Aug 23 2024

A066620 Number of unordered triples of distinct pairwise coprime divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 7, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 7, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 13, 0, 1, 2, 0, 1, 7, 0, 2, 1, 7, 0, 6, 0, 1, 2, 2, 1, 7, 0, 4, 0, 1, 0, 13, 1, 1, 1, 3, 0, 13, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 7, 0

Offset: 1

K. B. Subramaniam (kb_subramaniambalu(AT)yahoo.com) and Amarnath Murthy, Dec 24 2001

nonn

a(m) = a(n) if m and n have same factorization structure.

			a(24) = 3: the divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The triples are (1, 2, 3), (1, 2, 9), (1, 3, 4).
a(30) = 7: the triples are (1, 2, 3), (1, 2, 5), (1, 3, 5), (2, 3, 5), (1, 3, 10), (1, 5, 6), (1, 2, 15).

Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, vol. 12, No. 1-2-3, Spring 2001.pp 303-306.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of zeros are A000961.
Positions of ones are A006881.
The version for subsets of {1..n} instead of divisors is A015617.
The non-strict ordered version is A048785.
The version for pairs of divisors is A063647.
The non-strict version (3-multisets) is A100565.
The version for partitions is A220377 (non-strict: A307719).
A version for sets of divisors of any size is A225520.
A000005 counts divisors.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A018892 counts unordered pairs of coprime divisors (ordered: A048691).
A051026 counts pairwise indivisible subsets of {1..n}.
A337461 counts 3-part pairwise coprime compositions.
A338331 lists Heinz numbers of pairwise coprime partitions.
Cf. A007360, A023022, A084422, A276187, A282935, A305713, A337563, A337605, A343652, A343655.

Table[Length[Select[Subsets[Divisors[n],{3}],CoprimeQ@@#&]],{n,100}] (* Gus Wiseman, Apr 28 2021 *)

A066620(n) = (numdiv(n^3)-3*numdiv(n)+2)/6; \\ After Jovovic's formula. - Antti Karttunen, May 27 2017

from sympy import divisor_count as d
def a(n): return (d(n**3) - 3*d(n) + 2)/6 # Indranil Ghosh, May 27 2017

In the reference it is shown that if k is a squarefree number with r prime factors and m with (r+1) prime factors then a(m) = 4*a(k) + 2^k - 1.

a(n) = (tau(n^3)-3*tau(n)+2)/6. - Vladeta Jovovic, Nov 27 2004

More terms from Vladeta Jovovic, Apr 03 2003
Name corrected by Andrey Zabolotskiy, Dec 09 2020
Name corrected by Gus Wiseman, Apr 28 2021 (ordered version is 6*a(n))

A060277 Number of m for which a+b+c = n; abc = m has at least two distinct solutions (a,b,c) with 1 <= a <= b <= c.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 1, 0, 1, 1, 3, 1, 1, 1, 1, 3, 2, 7, 3, 2, 5, 4, 3, 5, 9, 2, 5, 6, 9, 5, 9, 14, 9, 7, 5, 10, 10, 11, 18, 7, 11, 16, 14, 12, 12, 23, 19, 13, 18, 11, 20, 19, 32, 17, 21, 18, 25, 19, 21, 27, 22, 21, 31, 27, 24, 28, 42, 34, 33, 21, 28, 31, 35, 47

Offset: 1

Naohiro Nomoto, Mar 23 2001

nonn

A triple (a,b,c) as described in the name cannot have c prime. - David A. Corneth, Aug 01 2018

			(14 = 6+6+2 = 8+3+3, 72 = 6*6*2 = 8*3*3); (14 = 8+5+1 = 10+2+2, 40 = 8*5*1 = 10*2*2); 14 has two "m" variables. so a(14)=2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1500 terms from Hugo Pfoertner)
IBM Research, Ponder This Challenge - July 2018.
John B. Kelly, Partitions with equal products, Proc. Amer. Math. Soc. 15 (1964), 987-990
John B. Kelly, Partitions with equal products. II, Proc. Amer. Math. Soc. 107 (1989), 887-893

Cf. A001399, A060275, A069905, A103277, A103278, A317388, A317578.

a[n_] := Count[ Tally[ Times @@@ IntegerPartitions[n, {3}]], {m_,c_} /; c>1]; Array[a, 84] (* Giovanni Resta, Jul 27 2018 *)

a(n)={my(M=Map()); for(i=n\3, n, for(j=(n-i+1)\2, min(n-1-i, i), my(k=n-i-j); my(m=i*j*k); my(z); mapput(M, m, if(mapisdefined(M, m, &z), z + 1, 1)))); #select(z->z>=2, if(#M, Mat(M)[, 2], []))} \\ Andrew Howroyd, Jul 27 2018

a(n) = Sum_{k>=2} A317578(n,k). - Alois P. Heinz, Aug 01 2018

Description revised by David W. Wilson and Don Reble, Jun 04 2002

cp's OEIS Frontend

A344294 5-smooth but not 3-smooth numbers k such that A056239(k) >= 2*A001222(k).

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A014591 a(n) = floor(n^2/12 + 5/4).

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A060022 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 3.

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A060023 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 4.

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Magma

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A325688 Number of length-3 compositions of n such that every distinct consecutive subsequence has a different sum.

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A325690 Number of length-3 integer partitions of n whose largest part is not the sum of the other two.

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A360010 First part of the n-th weakly decreasing triple of positive integers sorted lexicographically. Each n > 0 is repeated A000217(n) times.

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A375580 a(n) is the number of partitions n = x + y + z of positive integers such that x*y*z is a perfect cube.

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A375580 a(n) is the number of partitions n = x + y + z of positive integers such that xyz is a perfect cube.